Office_Shredder
Nov12-08, 05:31 PM
1. The problem statement, all variables and given/known data
Find the minimal polynomial of \frac{\sqrt{3}}{1+2^{1/3}} over Q
we'll call this x
2. Relevant equations
I wish I knew some :(
3. The attempt at a solution
By taking powers of x, I was able to show that the extension Q(x) has degree six (since 21/3 and sqrt(3) are both independently inside Q(x)) and hence the minimal polynomial has degree six. So then I took a general degree six polynomial, plugged in x, and got six equations in six unknowns. This is less than elegant, and I ended with a system of equations (luckily three of the six unknown coefficients were zero):
255b + 261d + 595f = -171
150b + 222d + 460f = -144
105b + 159d + 375f = -108
solving for b,d,f. An attempt at a numerical solution doesn't convince me this has a rational solution, but I wouldn't be surprised if there was a computational error preceding this. Is there a better way to do this?
Find the minimal polynomial of \frac{\sqrt{3}}{1+2^{1/3}} over Q
we'll call this x
2. Relevant equations
I wish I knew some :(
3. The attempt at a solution
By taking powers of x, I was able to show that the extension Q(x) has degree six (since 21/3 and sqrt(3) are both independently inside Q(x)) and hence the minimal polynomial has degree six. So then I took a general degree six polynomial, plugged in x, and got six equations in six unknowns. This is less than elegant, and I ended with a system of equations (luckily three of the six unknown coefficients were zero):
255b + 261d + 595f = -171
150b + 222d + 460f = -144
105b + 159d + 375f = -108
solving for b,d,f. An attempt at a numerical solution doesn't convince me this has a rational solution, but I wouldn't be surprised if there was a computational error preceding this. Is there a better way to do this?